Analysis Honors is the second half of a two year, assignment based course. GAT H is a prerequisite to this course.
Timeframe
Content
Essential Questions
Skills
Assessment/Resources
Assignments 60 through 67
Derivation and proof of polynomial formulas
- finite differences to derive polynomial formulas for sequences and geometric patterns
- mathematical induction as the way to prove that the derived formulas.
The student will:
- be able to use finite differences to derive polynomial formulas for sequences and geometric patterns
- be able to use mathematical induction to prove the derived formulas
questions on written tests
Assignments 60 through 63
Amortized Interest
(application of geometric series):
-derivation of amortized interest loan payment formula
-application of formula to real life loan situations
The student will
- understand the derivation of amortized interest loan payment formula
-apply formula to real life loan situations
questions on written tests
Assignments 60 through 83
Polynomials
- simple laws of exponents.
- investigation of cases of x-intercepts, turning points and end behavior (introduction to limit notation) by degree.
- connection between factored form and x-intercepts, and whether the x-intercepts are turning points.
- binomial expansion.
- multinomial expansion.
- writing the real coefficient equation (factored and simplified form) of a polynomial function given its intercepts/ zeros (includes complex conjugate theorem).
- adding, subtracting, multiplying and dividing ( both synthetic and long) polynomials.
- solving polynomial equations and inequalities, and systems of polynomial equations.
- relationship between roots of a polynomial and its simplified form coefficients.
How and why do polynomials behave as functions the way they do?
The student will be able to:
- describe cases of x-intercepts, turning points and end behavior by degree of polynomial.
- use limit notation to describe end behavior and asymptotic behavior/
- use factored form to describe x-intercepts, and whether the x-intercepts are turning points.
- write the real coefficient equation (factored and simplified form) of a polynomial function given its intercepts/ zeros (includes complex conjugate theorem).
- add, subtract, multiply and divide ( both synthetic and long) polynomials.
- solve polynomial equations and inequalities, and systems of polynomial equations.
- use the relationship between roots of a polynomial and its simplified form coefficients to quickly write polynomial expressions given its roots.
- create and analyze polynomial models for real-world situations.
- expand powers of binomials using the binomial theorem.
- expand powers, or at least find specific terms, of
multinomial expansions.
- simplifying operations with rational algebraic expressions.
- rational expressions with real exponents.
- rational functions (intercepts, vertical asymptotes, non-vertical - and some non-linear - asymptotes, holes.
- limit notation to describe end behavior and asymptotic behavior.
- writing the equation of rational functions given a description of their graph.
- applications of rational functions.
- decomposition of rational expressions into addend partial fractions.
- graphs of y = sec x , csc x, tan x and cot x (applying rational function properties).
The student shall be able to:
- simplify operations with rational algebraic expressions.
- simplify rational expressions with real exponents.
- sketch and/or describe the graph of rational functions (intercepts, vertical asymptotes, non-vertical - and some non-linear - asymptotes, holes.
- use limit notation to describe end behavior and asymptotic behavior.
- write the equation of rational functions given a description of their graph.
- solve applications of rational functions.
- decompose rational expressions into the sum of their partial fractions.
- graph y = sec x , y = csc x, y = tan x and y = cot x and transformations of them.
Pencil-and-paper tests and quizzes.
Assignments 69-87
Counting, Probability & Linear Programming
- definition of N!, and 0!
- combinations
- permutations of groups
- permutations with repeated elements
- permutations with replacement
- combinations with replacement
- probability definition.
- probability simulations.
- binomial probability.
- graphing linear inequalities (2-D)
- linear programming (application of linear inequalities)
- geometric probability.
The student will:
- understand and use the definition of N!, and 0!
- understand and use the correct counting method for combinations, permutations of groups, permutations with repeated elements, permutations with replacement, combinations with replacement
- understand the definition of probability.
- use simulations to approximate probabilities.
- recognize and compute binomial probabilities.
- recognize and compute geometric probabilities.
- graph linear inequalities (2-D).
- recognize and solve linear programming problems.
Questions on quizzes and tests.
Assignments 87-106
Exponentials, Inverses & Logarithms:
- function notation and concepts
- definition and application of function compositions
- logarithms as areas between x-axis and y = 1 / x.
- properties of logarithms from those area properties.
- definition of inverse function (function and geometric).
- inverse trig functions, their domains and ranges
- writing formulas of inverses of functions (restricting domains when necessary)
- the basic algebraic laws of exponents
- definition of exponential function, their domain and range
- exponential growth and decay
- definition of logarithmic function as inverse of exponential function, its domain and range
- statement, proof and application of the basic laws of logarithms
- natural and common logarithms
- the change-of-base property of logarithms
- solving exponential equations, equations involving logarithmic terms and systems involving logarithms
- half-life and other applications
The student -
- will understand and apply function notation and concepts
- can define and apply composition of functions
- will recognize logarithms as areas between x-axis and y = 1 / x.
- will understand and apply the definition of inverse function (both "function form" and geometric).
- will recognize and sketch the graph of inverse trig functions, and be able to describe their domains and ranges
- will write formulas of inverses of functions (restricting domains when necessary)
- will understand and apply the basic algebraic laws of exponents
- will understand and apply the definition of exponential function, and describe their domain and range
- will understand applications of exponential growth and decay
- will understand and apply the definition of logarithmic function as inverse of exponential function, and can describe its domain and range
- will understand, apply and be able to prove the basic laws of logarithms
- will understand natural and common logarithms
- will understand and apply the change-of-base property of logarithms
- will solve exponential equations, equations involving logarithmic terms and systems involving logarithms.
- will understand half-life and other applications
Questions on quizzes and tests
Assignments 88- 101
Volume & Surface Area
The formulation or derivation of formulas for the volume and surface of:
- cube
- right rectangular solid
- right pyramid
- right cone
- right cylinder
- sphere
- truncated cones
- composite solids
The student will:
- undersatnd the formulation or derivation of formulas for the volume and surface of all of the typical 3-D solids.
- apply the formulas to these figures and to solids composed of these basic solids.
Questions on quizzes and tests.
Assignments 84-89
Newton's Model
- explanation/ formulation of newton's model for freefalling motion (h=-.5gt^2+vt+h)
- application of formula in both displacement graphs and parametric representations of actual path.
The student-
- will understand, create and apply displacement graphs in freefalling body applications.
- will understand, create and apply parametric graphs in freefalling body applications.
Questions on quizzes and tests.
Assignments 90-106
Rates of Change/ Slope Generating Functions:
- creation of average rate of change expressions ( difference quotients ) in polynomial functions, sin x , cos x , e ^x and ln x.
- creation of expressions for the instantaneous rate of change for each of the given functions.
- writing equations of lines which are tangent to the given functions at particular values of x.
The student will:
- create average rate of change expressions
( difference quotients ) in polynomial functions, sin x , cos x , e ^x and ln x.
- create expressions for the instantaneous rate of change for each of the given functions.
- write equations of lines which are tangent to the given functions at particular values of x.
Questions on quizzes or tests
Assignment 98-115
Vector Mathematics
- definition of and notations for 2-D vectors
- definition of and formula for length/ magnitude of a 2-D vector
- addition and subtracting of 2-D vectors (geometric and formula)
- dot-product definition, derivation of cos theta for angle between two 2-D vectors
- extension of definitions/ derivations/ operations/ notations of all above for 3-D vectors.
- vector equations of lines in 3-D
- normal vector to a plane
- equations of planes in 3-D
- derivation for the projection of one vector onto another
- cross-products of 3-D vectors
- solving systems of 3-variable equations (plane intersections)
- finding intersections of 3-D lines and planes
The student -
- will understand and apply the definition of and notations for 2-D vectors
- will understand and apply the definition of and formula for length/ magnitude of a 2-D vector
- will add and subtract of 2-D vectors (geometric and formula)
- will understand and apply the dot-product definition, and use it to find the angle between two 2-D vectors
- will understand and apply the relationships when extending all of the definitions/ derivations/ operations/ notations of all above for 3-D vectors.
- will write vector equations of lines in 3-D
- will write equations of planes in 3-D (using normal vectors)
- will understand and apply the projection of one vector onto another
- will understand and apply the cross-products of 3-D vectors
- will solve systems of 3-variable equations (plane intersections)
- will find intersections of 3-D lines and planes, and angles between lines and/or planes
Questions on quizzes or tests
Assignments 100-117
Statistics
- definition and application of mean of a set
- definition and application of median of a set
- definition and application of mode of a set
- the five-point summary (quartiles), box-and-whisker plots
- frequency and bar graphs/ histograms
- definition and application of standard deviation
- transformations of data and their effect on the statistics of the set
- median-median line of fit for 2-var data set
- least squares regression line for 2-var data set
- coefficient of correlation for 2-var data set
The student:
- will understand and apply the definition and application of mean of a set
- will understand and apply the definition and application of median of a set
- will understand and apply the definition and application of mode of a set
- will understand and apply the five-point summary (quartiles), box-and-whisker plots
- will understand and apply frequency and bar graphs/ histograms
- will understand and apply the definition and application of standard deviation
- will understand and apply the transformation of data and its effect on the statistics of the set
- will understand and apply the median-median line of fit for 2-var data set
- lwill understand and apply the least squares regression line for 2-var data set
- will understand and apply the coefficient of correlation for 2-var data set
Questions on quizzes and tests
Assignments 115-124
Conic Sections
- geometric definitions of parabola, circle, ellipse and hyperbola (focus-directrix)
- symmetry of relations
- graphing conics from their equation, and identifying any vertices/ asymptotes, angles of rotation, etc.
- transformations of conics (translations, scale and size and rotations)- includes completing the square to change simplified equation into "transformed" equation
- writing rectangular equations of conics from their graph/ description
- writing parametric equations of conics from their graph/ description
- solving systems of equations involving conics
The student :
- will understand and apply the geometric definitions of parabola, circle, ellipse and hyperbola (focus-directrix)
- will understand and apply the symmetry of relations
- will graph conics from their equations (both rectangular and parametric), and identify any vertices/ asymptotes, angles of rotation, etc.
- graph/ describe transformations of conics (translations, scale and size and rotations)- includes completing the square to change simplified equation into "transformed" equation
- will write rectangular equations of conics from their graph/ description
- will write parametric equations of conics from their graph/ description
- will solve systems of equations involving conics
Questions on quizzes and tests
Assignment 122-127
Absolute Value & Other Piecewise Defined Functions
- definition of absolute value
- transformations of the graph of y = abs (x)
- peicewise defined functions (PWD )
- writing absolute value functions as PWD functions
The student will:
- understand and apply the definition of absolute value
- graph transformations of the graph of y = abs (x)
- understand and apply peicewise defined functions (PWD )
- write absolute value functions as PWD functions
Table of Contents
Derivation and proof of polynomial formulas
- finite differences to derive polynomial formulas for sequences and geometric patterns- mathematical induction as the way to prove that the derived formulas.
- be able to use finite differences to derive polynomial formulas for sequences and geometric patterns
- be able to use mathematical induction to prove the derived formulas
Amortized Interest
(application of geometric series):-derivation of amortized interest loan payment formula
-application of formula to real life loan situations
- understand the derivation of amortized interest loan payment formula
-apply formula to real life loan situations
Polynomials
- simple laws of exponents.- investigation of cases of x-intercepts, turning points and end behavior (introduction to limit notation) by degree.
- connection between factored form and x-intercepts, and whether the x-intercepts are turning points.
- binomial expansion.
- multinomial expansion.
- writing the real coefficient equation (factored and simplified form) of a polynomial function given its intercepts/ zeros (includes complex conjugate theorem).
- adding, subtracting, multiplying and dividing ( both synthetic and long) polynomials.
- solving polynomial equations and inequalities, and systems of polynomial equations.
- relationship between roots of a polynomial and its simplified form coefficients.
- describe cases of x-intercepts, turning points and end behavior by degree of polynomial.
- use limit notation to describe end behavior and asymptotic behavior/
- use factored form to describe x-intercepts, and whether the x-intercepts are turning points.
- write the real coefficient equation (factored and simplified form) of a polynomial function given its intercepts/ zeros (includes complex conjugate theorem).
- add, subtract, multiply and divide ( both synthetic and long) polynomials.
- solve polynomial equations and inequalities, and systems of polynomial equations.
- use the relationship between roots of a polynomial and its simplified form coefficients to quickly write polynomial expressions given its roots.
- create and analyze polynomial models for real-world situations.
- expand powers of binomials using the binomial theorem.
- expand powers, or at least find specific terms, of
multinomial expansions.
http://www.quickmath.com
Rational Expressions & Functions
- simplifying operations with rational algebraic expressions.- rational expressions with real exponents.
- rational functions (intercepts, vertical asymptotes, non-vertical - and some non-linear - asymptotes, holes.
- limit notation to describe end behavior and asymptotic behavior.
- writing the equation of rational functions given a description of their graph.
- applications of rational functions.
- decomposition of rational expressions into addend partial fractions.
- graphs of y = sec x , csc x, tan x and cot x (applying rational function properties).
- simplify operations with rational algebraic expressions.
- simplify rational expressions with real exponents.
- sketch and/or describe the graph of rational functions (intercepts, vertical asymptotes, non-vertical - and some non-linear - asymptotes, holes.
- use limit notation to describe end behavior and asymptotic behavior.
- write the equation of rational functions given a description of their graph.
- solve applications of rational functions.
- decompose rational expressions into the sum of their partial fractions.
- graph y = sec x , y = csc x, y = tan x and y = cot x and transformations of them.
Counting, Probability & Linear Programming
- definition of N!, and 0!- combinations
- permutations of groups
- permutations with repeated elements
- permutations with replacement
- combinations with replacement
- probability definition.
- probability simulations.
- binomial probability.
- graphing linear inequalities (2-D)
- linear programming (application of linear inequalities)
- geometric probability.
- understand and use the definition of N!, and 0!
- understand and use the correct counting method for combinations, permutations of groups, permutations with repeated elements, permutations with replacement, combinations with replacement
- understand the definition of probability.
- use simulations to approximate probabilities.
- recognize and compute binomial probabilities.
- recognize and compute geometric probabilities.
- graph linear inequalities (2-D).
- recognize and solve linear programming problems.
Exponentials, Inverses & Logarithms:
- function notation and concepts- definition and application of function compositions
- logarithms as areas between x-axis and y = 1 / x.
- properties of logarithms from those area properties.
- definition of inverse function (function and geometric).
- inverse trig functions, their domains and ranges
- writing formulas of inverses of functions (restricting domains when necessary)
- the basic algebraic laws of exponents
- definition of exponential function, their domain and range
- exponential growth and decay
- definition of logarithmic function as inverse of exponential function, its domain and range
- statement, proof and application of the basic laws of logarithms
- natural and common logarithms
- the change-of-base property of logarithms
- solving exponential equations, equations involving logarithmic terms and systems involving logarithms
- half-life and other applications
- will understand and apply function notation and concepts
- can define and apply composition of functions
- will recognize logarithms as areas between x-axis and y = 1 / x.
- will understand and apply the definition of inverse function (both "function form" and geometric).
- will recognize and sketch the graph of inverse trig functions, and be able to describe their domains and ranges
- will write formulas of inverses of functions (restricting domains when necessary)
- will understand and apply the basic algebraic laws of exponents
- will understand and apply the definition of exponential function, and describe their domain and range
- will understand applications of exponential growth and decay
- will understand and apply the definition of logarithmic function as inverse of exponential function, and can describe its domain and range
- will understand, apply and be able to prove the basic laws of logarithms
- will understand natural and common logarithms
- will understand and apply the change-of-base property of logarithms
- will solve exponential equations, equations involving logarithmic terms and systems involving logarithms.
- will understand half-life and other applications
Volume & Surface Area
The formulation or derivation of formulas for the volume and surface of:- cube
- right rectangular solid
- right pyramid
- right cone
- right cylinder
- sphere
- truncated cones
- composite solids
- undersatnd the formulation or derivation of formulas for the volume and surface of all of the typical 3-D solids.
- apply the formulas to these figures and to solids composed of these basic solids.
Newton's Model
- explanation/ formulation of newton's model for freefalling motion (h=-.5gt^2+vt+h)- application of formula in both displacement graphs and parametric representations of actual path.
- will understand, create and apply displacement graphs in freefalling body applications.
- will understand, create and apply parametric graphs in freefalling body applications.
Rates of Change/ Slope Generating Functions:
- creation of average rate of change expressions ( difference quotients ) in polynomial functions, sin x , cos x , e ^x and ln x.- creation of expressions for the instantaneous rate of change for each of the given functions.
- writing equations of lines which are tangent to the given functions at particular values of x.
- create average rate of change expressions
( difference quotients ) in polynomial functions, sin x , cos x , e ^x and ln x.
- create expressions for the instantaneous rate of change for each of the given functions.
- write equations of lines which are tangent to the given functions at particular values of x.
Vector Mathematics
- definition of and notations for 2-D vectors- definition of and formula for length/ magnitude of a 2-D vector
- addition and subtracting of 2-D vectors (geometric and formula)
- dot-product definition, derivation of cos theta for angle between two 2-D vectors
- extension of definitions/ derivations/ operations/ notations of all above for 3-D vectors.
- vector equations of lines in 3-D
- normal vector to a plane
- equations of planes in 3-D
- derivation for the projection of one vector onto another
- cross-products of 3-D vectors
- solving systems of 3-variable equations (plane intersections)
- finding intersections of 3-D lines and planes
- will understand and apply the definition of and notations for 2-D vectors
- will understand and apply the definition of and formula for length/ magnitude of a 2-D vector
- will add and subtract of 2-D vectors (geometric and formula)
- will understand and apply the dot-product definition, and use it to find the angle between two 2-D vectors
- will understand and apply the relationships when extending all of the definitions/ derivations/ operations/ notations of all above for 3-D vectors.
- will write vector equations of lines in 3-D
- will write equations of planes in 3-D (using normal vectors)
- will understand and apply the projection of one vector onto another
- will understand and apply the cross-products of 3-D vectors
- will solve systems of 3-variable equations (plane intersections)
- will find intersections of 3-D lines and planes, and angles between lines and/or planes
Statistics
- definition and application of mean of a set- definition and application of median of a set
- definition and application of mode of a set
- the five-point summary (quartiles), box-and-whisker plots
- frequency and bar graphs/ histograms
- definition and application of standard deviation
- transformations of data and their effect on the statistics of the set
- median-median line of fit for 2-var data set
- least squares regression line for 2-var data set
- coefficient of correlation for 2-var data set
- will understand and apply the definition and application of mean of a set
- will understand and apply the definition and application of median of a set
- will understand and apply the definition and application of mode of a set
- will understand and apply the five-point summary (quartiles), box-and-whisker plots
- will understand and apply frequency and bar graphs/ histograms
- will understand and apply the definition and application of standard deviation
- will understand and apply the transformation of data and its effect on the statistics of the set
- will understand and apply the median-median line of fit for 2-var data set
- lwill understand and apply the least squares regression line for 2-var data set
- will understand and apply the coefficient of correlation for 2-var data set
Conic Sections
- geometric definitions of parabola, circle, ellipse and hyperbola (focus-directrix)- symmetry of relations
- graphing conics from their equation, and identifying any vertices/ asymptotes, angles of rotation, etc.
- transformations of conics (translations, scale and size and rotations)- includes completing the square to change simplified equation into "transformed" equation
- writing rectangular equations of conics from their graph/ description
- writing parametric equations of conics from their graph/ description
- solving systems of equations involving conics
- will understand and apply the geometric definitions of parabola, circle, ellipse and hyperbola (focus-directrix)
- will understand and apply the symmetry of relations
- will graph conics from their equations (both rectangular and parametric), and identify any vertices/ asymptotes, angles of rotation, etc.
- graph/ describe transformations of conics (translations, scale and size and rotations)- includes completing the square to change simplified equation into "transformed" equation
- will write rectangular equations of conics from their graph/ description
- will write parametric equations of conics from their graph/ description
- will solve systems of equations involving conics
Absolute Value & Other Piecewise Defined Functions
- definition of absolute value- transformations of the graph of y = abs (x)
- peicewise defined functions (PWD )
- writing absolute value functions as PWD functions
- understand and apply the definition of absolute value
- graph transformations of the graph of y = abs (x)
- understand and apply peicewise defined functions (PWD )
- write absolute value functions as PWD functions